The World of Numbers (RBSE Class 9 · Mathematics)

Long before cities or laws, humans scratched tally marks on bone to count. India then gave the world its most powerful idea — śhūnya (zero) — and the place-value system we still use. This chapter walks that journey and then builds the full real number system.

RBSE note (2026-27). Class 9 uses the new NCF (Ganita Prakash 9) Mathematics textbook. The World of Numbers is the new book's number-system chapter (it opens with the history of counting and zero). BSER (Ajmer) sets the exam.

1. The story so far — types of numbers

The Indian invention of śhūnya (zero) and Brahmagupta's rules for operating with it made our place-value arithmetic possible.

2. Rational numbers

A rational number can be written as p/q with integers p, q and q ≠ 0. Key facts:

• Between any two rational numbers there are infinitely many rationals (e.g. the average lies between them).
• The decimal form of a rational number is either terminating (0.75) or non-terminating but recurring (0.333… = ⅓).

3. Irrational numbers

An irrational number cannot be expressed as p/q. Its decimal expansion is non-terminating and non-recurring. Examples: √2, √3, π, and 0.101001000100001…

√2 is irrational — a famous proof by contradiction shows no fraction squares to exactly 2.

Together, the rationals and irrationals fill the entire number line — they are the real numbers, and every point on the number line is exactly one real number.

4. Representing numbers on the number line

• Any rational number can be marked by successive division of unit segments.
• Irrationals like √2 are located using the Pythagoras construction: a right triangle with legs 1 and 1 has hypotenuse √2; sweep that length onto the line with a compass. Repeating gives the square-root spiral.

5. Operations and laws of exponents for real numbers

For real numbers and rational exponents (a, b > 0):

$a^m\cdot a^n=a^{m+n}\frac{a^m}{a^n}=a^{m-n}(a^m{)}^n=a^{mn}$ $a^m{b}^m=(ab{)}^m{a}^{1/n}=\sqrt[n]{a}$

Surds (roots of non-perfect numbers) follow: √a · √b = √(ab) and √a / √b = √(a/b).

6. Rationalising the denominator

To remove a surd from a denominator, multiply by a clever form of 1:

$\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{a}}\times \frac{\sqrt{a}}{\sqrt{a}}=\frac{\sqrt{a}}{a}$

For a denominator like (√a + √b), multiply by its conjugate (√a − √b):

$\frac{1}{\sqrt{a}+\sqrt{b}}\times \frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{\sqrt{a}-\sqrt{b}}{a-b}$

7. Worked example

Simplify $\frac{1}{\sqrt{3}-\sqrt{2}}$ by rationalising the denominator.

Multiply numerator and denominator by the conjugate (√3 + √2):

$\frac{1}{\sqrt{3}-\sqrt{2}}\times \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}=\frac{\sqrt{3}+\sqrt{2}}{(\sqrt{3}{)}^2-(\sqrt{2}{)}^2}=\frac{\sqrt{3}+\sqrt{2}}{3-2}=\sqrt{3}+\sqrt{2}.$

8. Quick recap

• ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ; irrationals + rationals = real numbers (fill the number line).
• Rational = p/q (q ≠ 0); decimals terminate or recur. Irrational = non-terminating, non-recurring.
• Locate irrationals on the line by the Pythagoras/√-spiral construction.
• Use the laws of exponents and surd rules to simplify.
• Rationalise denominators using the conjugate; (√a)² − (√b)² = a − b.